SINGLE OUTPUT DEPENDENT QUADRATIC OBSERVABILITY NORMAL FORM G Zheng D Boutat JP Barbot INRIA Rhône-Alpes, Inovallée, 655 avenue de l Europe, Montbonnot Sant Martn, 38334 St Ismer Cedex, France LVR/ENSI, Boulevard de Lahtolle, 8 Bourges, France ECS/ENSEA, 6 Avenue du Ponceau, 954 Cergy-Pontose, France, Projet ALIEN-INRIA Futures Abstract: In ths paper, two quadratc observablty normal forms dependent on output respectvely correspondent to drft domnant term forced domnant term are frstly dscussed Moreover characterstc numbers are studed n order to smplfy the calculaton of these two normal forms INTRODUCTION Normal form s a powerful tool to analyze the propertes of dynamcal systems, such as stablty (Poncaré et al, 899), controllablty (Kang et al, 99) In, ths concept was frstly ntroduced n (Boutat-Baddas et al, ) n order to analyze the observablty of dynamcal system, n whch the quadratc observablty normal form was studed the nterests of ths technque was hghlghted In 5, a new observablty normal form dependent on ts output was studed n (Zheng et al, 5), the necessary suffcent condtons were proposed Moreover, an extenson to mult-outputs case has also been analyzed n (Boutat et al, 6) Snce the observablty normal form dependent on ts output (or outputs) were exactly lnear, another reasonable extenson based on the work of (Boutat-Baddas et al, ) (Chabraou et al, 3) s the study of quadratc observablty normal form dependent on ts output (or outputs) Therefore, n ths paper, we consder the followng sngle nput sngle output system: ζ D IR n, u IR, f : IR n IR n, g : IR n IR n h : IR n IR are analytc functons, assume that for all ζ D, we have rank [ dh, dl f h,, dl f h T = n As an extenson of our work n (Zheng et al, 5), f the theorem n (Zheng et al, 5) s not verfed, we assume that system () can be transformed nto the followng form : η = β(y) A(y)η γ y [ ( η) B(y)u ϑ y ( η)u O y [3 ( η, u) y = Cη η IR n, η = [η, η T, γ [ y ( η) = [γ [ y ( η),, γ [ y n ( η) T, ϑ y ( η) = [ϑ y ( η),, ϑ y n ( η) T, for n, γ y [ ( η) ϑ y ( η) are functon of η wth order respectvely parameterzed by y, β(y) = [β (y),, β n(y) T, α (y) A(y) =, α (y) () ζ = f(ζ) g(ζ)u y = h(ζ) () The necessary suffcent condton for ths transformaton s stll an open queston
B(y) s defned as follows: B(y) = ( b(y),,, ) T (3) Assume that z = x φ [ y, hence ż = ẋ φ[ Accordng to equaton (4) (5), we have y ẋ x As t was shown n (Boutat-Baddas et al, ) that the equvalence modulo an output njecton s justfed by the fact that the output njecton can be canceled n the observaton error dynamcs Therefore, by an output njecton β(y), system () s equvalent modulo an output njecton to the followng system ẋ = A(y)x γ y [ B(y)u ϑ y u O y [3 ( x, u) (4) y = x n = Cx x = [x,, x T Our problem s how to characterze the fact that all the quadratc terms can be canceled by a dffeomorphsm? If ths knd of dffeomorphsm does not exst, then what s ts normal form ts resonant terms? QUADRATIC EQUIVALENCE MODULO AN OUTPUT INJECTION Defnton System (4) s quadratcally equvalent to system ż = A(y)z γ y [ ( z) B(y)u y ( z)u O y [3 ( z, u) (5) y = Cz f there exsts a dffeomorphsm of the followng form: z = x φ [ y (6) whch transforms the quadratc term γ y [ nto another quadratc term γ y [ ( z), φ [ y = [φ [ y,,, φ[ y,n T φ [ y, are the homogenous polynomals wth order n z Remark ) In order to keep the output unchanged, we choose the output equal to x n, whch means the dffeomorphsm z = x φ [ y should verfy φ [ = ) It should be noted that ths choce s not oblgatory In fact, we can also choose φ [ = φ [ (y), e, a functon of the output Proposton System (4) s quadratcally equvalent to system (5) modulo an output njecton, f only f the followng homologc equatons are satsfed: γ y [ Γ [ y = A(y)φ [ y γ y [ ϑ y φ[ y b(y) = x Γ [ [ y = φ [ y x α (y),, y φ [ y α n (y),, x x (7) ż = [ φ[ y x ( = A(y) x φ [ y B(y)u [ A(y)x γ y [ B(y)u ϑ y u O y [3 ( x, u) ) γ y [ ( x φ [ y ) y u O [3 y ( x, u) So we obtan γ y [ φ[ y x ϑ y φ[ y B(y) = x A(y)x = A(y)φ [ y γ [ y y φ [ y A(y)x [ x φ [ = y φ [ y α (y),, x y = Γ [ y O y [3 α (y),, fnally equaton (8) becomes: γ y [ Γ [ y = A(y)φ [ y γ y [ ϑ y φ[ y b(y) = y x 3 QUADRATIC OBSERVABILITY NORMAL FORM DEPENDENT ON ITS OUTPUT Snce there exst two quadratc terms n system (5): γ y [ ( z) y ( z)u, we wll study two normal form whch correspond respectvely to the drft term forced term n ths secton 3 Normal form wth drft domnant term In ths subsecton, we study the normal form correspondent to drft domnant term by smplfyng the quadratc term γ y [ ( z) Theorem Normal form correspondent to drft domnant term system (5) by quadratcally equvalent modulo an output njecton s n the followng normal form: c (y)ξ h j (y)ξ ξ j j ξ = u c (y)ξ c n (y)ξ x (8) A(y)ξ B(y)u O y [3 ( ξ, u) (9)
Snce the objectve of ths normal form s to make γ y [ =, so the frst homologc equaton n (7) becomes Defne γ [ y Γ [ y = A(y)φ [ y φ [ y = [φ [,, φ [ T, γ y [ = [γ y, [,, γ y, [ n T, ϑ y = [ ϑ,, ϑ T, φ [ y n = And we obtan n α (y)φ [ = γ y, [ [ φ y, α (y)x x α n (y)φ [ [ = γ y, [ n φ [ α (y)x x α (y)φ [ = γ y, [ n () the frst lne of equaton (7) gves n γ y, [ [ φ x α (y) = () x Equaton () can be used to deduce φ [ y, n order to cancel the quadratc terms from γ y, [ to γ y, [ n respectvely Moreover, f γ y, [ φ [ y, verfy also equaton (), then ths system can be quadratcally lnearzable Otherwse, t gves the followng resonant terms: Wth ϑ n γ y, [ α (y) φ[ y, x x y φ[ y ϑ Assume b(y) = x y, we have = = c n (y)x ϑ = c n (y)x, then c n (y) = cn (y) And t s not possble to cancel other components Hence we obtan normal form (9) In order to hghlght the proposed method, we consder the followng example Example Consder the followng system: ( ) ẋ = x ( x 3 x x x 3 x x 3 4x 3 x x 4 3 ) ( x u ẋ = x 3 x x 3 x x 3 x x 3 ) x u ẋ 3 = x 3 x x 3 x u y = x 3 () y Accordng to equaton (), we have φ [ y,3 = α (y)φ [ y, = γ y [,3 α (y)φ [ = γ y [, α (y) φ[ y, x x } z = x x 3 x we obtan z = x Hence the quadratc z 3 = x 3 resonant terms are: γ y, [ α (y) φ[ x = x x x 3 x Fnally we have the followng normal form: ( ż = z z 3 z z 3 4z3 z z3 ) 4 z u O y [3 (z (, z, u) ) ż = z 3 z z 3 z z3 z [3 u O y (z, z, u) ż 3 = z 3 z z3 z u O y [3 (z, z, u) y = z 3 3 Normal form wth forced domnant term (3) Ths subsecton s devoted to study another normal form by smplfyng quadratc terms: y ( z)u, n system (5) Theorem Normal form forced domnant term of system (5) by quadratcally equvalent modulo an output njecton s n the followng form: d (y)ξ ξ j j ξ d ξ = (y)ξ u c n (y)ξ (4) ξ d n (y)ξ A(y)ξ B(y)u O y [3 ( ξ, u) Assumng φ [ =, we have f we set then we obtan ϑ = ϑ y φ[ y b(y) = (5) x = = = Accordng to the frst homologc equaton n (7), γ [ y Γ [ y = A(y)φ [ y γ [ y (6) we obtan γ y, [ = γ y, [ γ [ y, α (y)φ [ n [ φ x α (y) x = γ y, [ n γ y, [ α n (y)φ [ [ φ [ y, x α (y)x [ = γ y, [ n φ [ α (y)x x γ [ α (y)φ [ = γ [ From equaton (5), the term φ [ y can be used to cancel all the quadratc terms form the second lne
to the last one, except for the terms x d j x, j [, n, hence we obtan: j γ y [ = ( d (y)x x j, x d (y)x,, x d n (y)x Fnally we have the normal form (4) The followng example s to llustrate the proposed normal form Example (Example contnue) Accordng to the above method, by smple calculaton, we have z = x x 3 x x3 3 x x x 3 x z = x x x x 3 x (7) z 3 = x 3 wth ths dffeomorphsm we have the followng normal form: ( ż = z3) 4 z z3 z z 3u O y [3 ( (z, z, u) ż = z 3 z z 3 z3) z z3 4 z z O y [3 (z, z, u) (8) ż 3 = z 3 z z 3 z z z3 z z 3 z u O y [3 (z, z, u) y = z 3 4 CHARACTERISTIC NUMBERS In order to smplfy the calculaton of dffeomorphsm proposed before, a new method wll be proposed n ths secton whch permt us to determne the dffeomorphsm (6) n a easer way, wth whch we need not to solve the homologc equaton (7) 4 Characterstc numbers for normal form wth drft domnant term In order to smplfy equaton (7), we assume: φ [ y, = x T φ y, x, γ y, [ = x T γ y, x, γ y, [ = x T γ y, x, φ φ y, := (y) φ, (y) φ, (y) φ, (y), γ (y) γ, (y) γ y, := γ, (y) γ, (y), γ (y) γ, (y) γ y, := γ, (y) γ, (y) ) T We obtan Γ [ y = [Γ [,, Γ [ T Γ [ y, = x T Γ y, x Γ y, = ĀT (y)φ y, φ y, Ā(y), α (y) Ā(y) := α n (y) Settng A(y)φ [ y := x T φ yx, for all x, homologc equaton (7) can be wrtten as follows: And we have x T γ yx x T Γ yx = x T φyx x T γ yx Because A(y)φ [ y = γ y Γ y = φ y γ y (9) f γ y =, we can cancel all the quadratc terms n system (4) α (y)φ [ f γ y [ =, then α (y)φ [ φ y, γ y, φ y, γ y, φ y, φ y, ĀT (y) γ y,n ( φ y,n φ y,n ) T =, α (y)φ y,,, α (y)φ y, Ā(y) we have γ y, ĀT (y)φ y, φ y, Ā(y) = α (y)φ y, = γ y, ĀT (y)φ y, φ y, Ā(y), for n Fnally, by recurrence, we get k [ C k (ĀT ) k j j (y) (y) γy,kāj j= φ y, = k k= α m m= () for n, Cj k denotes the combnatoral coeffcent Wth ths dffeomorphsm, accordng to the followng equalty: ϑ y φ[ b(y) = x ϑ y,n = Settng then c n j (y) = cn j (y) y ϑ = c n j (y)x j, = j= c n j (y)x j, j= y we have
Because ϑ y φ[ b(y) = x y ϑ y, = c j (y)x j, y, = j= y, assumng c j (y)x j j= then, snce φ [ y, = x T φ y, x, we obtan: c j (y) = c j (y) b(y)φ,j (y), c n j (y) = cn j (y) () for, j n Defnton We defne the characterstc matrx for system (4) as follows: M y = γ y, ĀT (y)φ y, φ y, Ā(y) () c (y) c (y) C y = (3) c n (y) cn (y) M y C y are only functon of y We are now able to set the followng theorem Theorem 3 Normal form correspondent to drft domnant term of system (5) s as follows: M y ξ = ξ T ξ A(y)ξ B(y)u C yξu O [3 y ( ξ, u) M y s defned n () (4) Remark ) M y(, j) n equaton (4) depends on coeffcents h j (y) of equaton (9) as follows: M y(, j) = M y(j, ) = h j(y), < j M y(, ) = h (y) ) there are 3n(n )/ characterstc numbers n the normal form Example 3 (Example contnue) Wth the gude of the computaton process proposed above, we have ( ) M y = γ y, ĀT (y)φ y, φ y, Ā(y) = x 3 ( ) 4x 3 x 4 3 C y = x 3 x 3 Fnally we get the x 3 same normal form correspondent to drft domnant term n (3) 4 Characterstc numbers for normal form wth forced domnant term Wth the same argument, n order to smplfy equaton (7), settng ϑ = c n j (y)x j, = j= c n j (y)x j, j= c n j (y) = cn j (y) (5) Snce ϑ y φ[ b(y) = x y, φ (y) φ, (y) φ [ y, = x T φ, (y) φ, (y) x, y f we set φ,j (y) = c j (y), we obtan b(y) = = = Then we have γ y Γ y = φ y γ y, whch gves γ y, Ā(y)φy, φā(y) = γ γ y, α (y)φ y, = γ y, ĀT (y)φ y, φ y, Ā(y) for n Defnng Υ (y) Υ, (y) Υ y, = Υ, (y) Υ, (y) = [ k C k k j γ y,k j (y) Ā γ (y) j= y,k k= k α m m= (6) Ῡ (y) Ῡ, (y) Ῡ y, = (7) Ῡ, (y) Ῡ, (y) k [ C k k j j (y) (y) γy,kāj j= = k k= α m m= Because φ,j (y) = c j (y),, j [, n, f we set b(y) φ l,s (y) = Ῡ l,s (y), [, n, l, s [, n we have γ l,s =, [, n, l, s [, n Hence we get the followng dffeomorphsm: φ φ y, :,j (y) = c j (y),, j [, n b(y) φ l,s (y) = Ῡ l,s (y), [, n, l, s [, n
Because we have γ l,s =, [, n, l, s [, n, then γ,j, [, n, j [, n can be determned by the followng equaton: we note φ,j (y) = c j (y) b(y) = Υ,j (y) M yk (, j) = M yk (j, ) = for j n k n γ k,j, =,, n, (8) Fnally because γ y, Ā(y)φ φā(y) = γy,, we obtan: M y, (, j) = M y, (j, ) = γ,j, < j (9) Then we can gve the followng theorem Theorem 4 Normal form correspondent to forced domnant term of system (5) s as follows: M y, M y, ξ = ξ T ξ u M y,n c n (y)ξ A(y)ξ B(y)u O y [3 ( ξ, u) c n M y, are defned by equatons (5), (8) (9) Remark 3 The number of the free coeffcents s 3n(n )/ Example 4 (Example contnue) Followng the proposed calculaton procedure, we can obtan x3 x 3 3 x 3 φ (y) = x 3 3 x 3, φ (y) =, φ 3 (y) = the followng matrx M y: ( ) x 4 3 M y, = x 3, M y, = x 3 x 3 M y,3 = x 3 whch yelds the same normal form (8) ( ) x3 x 3 x4 3 x 4 3, studed two normal forms respectvely correspondent to the drft domnant term forced domnant term Both representatons are equvalent In order to smplfy the calculaton, we proposed to apply quadratc terms characterstc matrx for these two normal forms REFERENCES Boutat D, Zheng G, Barbot JP Hammour H (6), Observer Error Lnearzaton Mult-Output Dependng, In Proc of IEEE CDC, 6 Boutat-Baddas L, Boutat D, Barbot JP Taulegne R(), Quadratc Observablty normal form, In Proc of IEEE CDC, Chabraou S, Boutat D, Boutat-Baddas L Barbot JP (3), Observablty quadratc characterstc numbers, In Proc of IEEE CDC, 3 Poncaré H (899), Les Méthodes nouvelles de la mécanque céleste, Gauther Vllard, 899 Réedton 987, Bbblothèque scentfque A Blanchard Krener AJ, (984), Approxmate lnearzaton by state feedback coordnate change, Systems Control Letter, 5, 8-85 Krener AJ, Karahan S, Hubbard M Frezza R (987), Hgher order lnear approxmatons to nonlnear control systems, In Proc of IEEE CDC, 987 Krener AJ, Karahan S Hubbard M, (988), Approxmate normal forms of nonlnear systems, In Proc of IEEE CDC, 988 Kang W Krener AJ (99), Extended quadratc controller normal form dynamc state feedback lnearzaton of non lnear systems, SIAM J Control Optmzaton, Vol 3, No 6, pp 39-337, 99 Zheng G, Boutat D Barbot JP (5), Output Dependent Observablty Normal Form, In Proc of IEEE CDC, 5 5 CONCLUSION Ths paper s devoted to study the quadratc observablty normal form parameterzed by ts output Above all, two homologc equatons were gven n order to guarantee the equvalence of quadratc transformaton After that, we have